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Thermodynamics Of Some Nonuniformly Hyperbolic Dynamical Systems
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Book Synopsis Thermodynamics of Some Nonuniformly Hyperbolic Dynamical Systems by : Dominic Veconi
Download or read book Thermodynamics of Some Nonuniformly Hyperbolic Dynamical Systems written by Dominic Veconi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the primary tools used to study hyperbolic and expanding dynamical systems are equilibrium measures. These are probability measures that are invariant under the dynamical system, and which maximize the sum of the metric entropy of the dynamics and the integral of a particular function (this maximum is taken over all invariant probability measures). Within the equilibrium measures are the Sinai-Ruelle-Bowen (SRB) measures. These are equilibrium measures [mu] that are "physical" in the sense that for Lebesgue-almost every point x in the phase space X and every continuous real-valued function [phi] on X , the Birkhoff time average is equal to the space of the function [phi]. It is known that if a surface diffeomorphism admits an SRB measure, then this measure is unique. Furthermore, hyperbolic maps admitting singularities have at most countably many ergodic SRB measures. In this dissertation, we consider the SRB measures and thermodynamic properties of three classes of dynamical systems: almost Anosov toral diffeomorphisms; smooth realizations of pseudo-Anosov maps; and singular hyperbolic attractors. Almost Anosov and pseudo-Anosov maps are surface diffeomorphisms admitting a finite number of singular fixed points, where the differential of the map slows to the identity. We develop a thermodynamic formalism for both classes of diffeomorphisms. In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium measures for a collection of non-Hölder continuous geometric potentials. For both almost and pseudo-Anosov maps, this family of equilibrium measures includes a unique SRB measure and a measure of maximal entropy. This measure of maximal entropy also has exponential decay of correlations and satisfies the Central Limit Theorem. Singular hyperbolic attractors are the invariant supports of physical measures for dissipative hyperbolic systems. Examples of singular hyperbolic attractors include the geometric Lorenz attractor (as well as related Lorenz-type systems), and the Belykh attractor. We establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary.
Book Synopsis THERMODYNAMIC FORMALISM BEYOND UNIFORMLY HYPERBOLIC SYSTEMS. by : Agnieszka Zelerowicz
Download or read book THERMODYNAMIC FORMALISM BEYOND UNIFORMLY HYPERBOLIC SYSTEMS. written by Agnieszka Zelerowicz and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis is to study thermodynamics of dierentiable dynamical systems which display various forms of hyperbolicity. In the classical, uniformly hyperbolic setting we propose a new way of constructing equilibrium measures which uses some tools from geometric measure theory. We then use it to construct equilibrium measures for a certain class of partially hyperbolic systems. Almost immediately from the construction one obtains that those measures are unique equilibrium measures with full support which have local product structure. We also consider systems which are nonuniformly hyperbolic We present a systematic study of thermodynamics formalism by methods of inducing. We give a complete overview of known results and prove additional ones. One of the main contributions is the proof of Bernoulli property for equilibrium measures corresponding to a certain class of potential functions. We also prove analyticity of the pressure function for a certain class of families of potential functions. Finally,we study a certain class of nonuniformly hyperbolic attractors. Those are attractors obtained from uniformly hyperbolic systems by the slow down procedure. We study the family of geometric t-potentials on some interval in t. We show existence and uniqueness of equilibrium measures for potentials in this family. We prove that those equilibrium measures satisfy the Central Limit Theorem, have exponential decay of correlations, and the Bernoulli property. Finally, we prove that the pressure function is real analytic.
Book Synopsis Thermodynamic Formalism by : Mark Pollicott
Download or read book Thermodynamic Formalism written by Mark Pollicott and published by Springer Nature. This book was released on 2021-10-01 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism: Applications to Probability, Geometry and Fractals” which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.
Book Synopsis Thermodynamics by : Wassim M. Haddad
Download or read book Thermodynamics written by Wassim M. Haddad and published by Princeton University Press. This book was released on 2009-01-10 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics. The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth. This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.
Book Synopsis Thermodynamic Formalism and Holomorphic Dynamical Systems by : Michel Zinsmeister
Download or read book Thermodynamic Formalism and Holomorphic Dynamical Systems written by Michel Zinsmeister and published by American Mathematical Soc.. This book was released on 2000 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of thermodynamics and statistical physics is to understand the equilibrium of a gas or the different states of matter. To understand the strange fractal sets appearing when one iterates a quadratic polynomial is one of the goals of the theory of holomorphic dynamical systems. These two theories are strongly linked: The laws of thermodynamics happen to be an extremely powerful tool for understanding the objects of holomorphic dynamical systems. A "thermodynamic formalism" has been developed, bringing together notions that are a priori unrelated. While the deep reasons of this parallelism remain unknown, the goal of this book is to describe this formalism both from the physical and mathematical point of view in order to understand how it works and how useful it can be. This translation is a slightly revised version of the original French edition. The main changes are in Chapters 5 and 6 and consist of clarification of some proofs and a new presentation of the basics in iteration of polynomials.
Book Synopsis Hyperbolic Dynamics, Fluctuations and Large Deviations by : D. Dolgopyat
Download or read book Hyperbolic Dynamics, Fluctuations and Large Deviations written by D. Dolgopyat and published by American Mathematical Soc.. This book was released on 2015-04-01 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the semester-long special program on Hyperbolic Dynamics, Large Deviations and Fluctuations, which was held from January-June 2013, at the Centre Interfacultaire Bernoulli, École Polytechnique Fédérale de Lausanne, Switzerland. The broad theme of the program was the long-term behavior of dynamical systems and their statistical behavior. During the last 50 years, the statistical properties of dynamical systems of many different types have been the subject of extensive study in statistical mechanics and thermodynamics, ergodic and probability theories, and some areas of mathematical physics. The results of this study have had a profound effect on many different areas in mathematics, physics, engineering and biology. The papers in this volume cover topics in large deviations and thermodynamics formalism and limit theorems for dynamic systems. The material presented is primarily directed at researchers and graduate students in the very broad area of dynamical systems and ergodic theory, but will also be of interest to researchers in related areas such as statistical physics, spectral theory and some aspects of number theory and geometry.
Book Synopsis Decay of Correlations for Dynamical Systems with Non-zero Lyapunov Exponents by : Farruh Shahidi
Download or read book Decay of Correlations for Dynamical Systems with Non-zero Lyapunov Exponents written by Farruh Shahidi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis is to study thermodynamic formalism as well as ergodic properties of equilibrium measures for certain classes of non-uniformly hyperbolic systems. We present known results which use the inducing method and also prove additional results including the Bernoulli property, decay of correlations and the Central Limit Theorem (CLT) for equilibrium measures. We also prove the analyticity of the pressure function for some class of potential functions. We then apply these results to the Katok map on the two-dimensional torus to prove that the map has a polynomial(upper and lower) decay of correlations and satisfies the CLT with respect to the area. Using the Katok map we then construct an area preserving surface diffeomorphism of C1+[beta] regularity that has polynomial decay of correlations with respect to the Lebesgue measure.
Book Synopsis Thermodynamics of Chaos and Order by : V Berdichevsky
Download or read book Thermodynamics of Chaos and Order written by V Berdichevsky and published by CRC Press. This book was released on 1997-10-24 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: The discovery of chaotic motion in low-dimensional systems raised the question: What kind of thermodynamics describes a system if it is neither ergodic nor Hamiltonian or possesses a finite number of degrees of freedom? This Monographs is the first to discuss this question.
Book Synopsis Nonuniformly Hyperbolic Attractors by : José F. Alves
Download or read book Nonuniformly Hyperbolic Attractors written by José F. Alves and published by Springer Nature. This book was released on 2020-12-19 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph offers a coherent, self-contained account of the theory of Sinai–Ruelle–Bowen measures and decay of correlations for nonuniformly hyperbolic dynamical systems. A central topic in the statistical theory of dynamical systems, the book in particular provides a detailed exposition of the theory developed by L.-S. Young for systems admitting induced maps with certain analytic and geometric properties. After a brief introduction and preliminary results, Chapters 3, 4, 6 and 7 provide essentially the same pattern of results in increasingly interesting and complicated settings. Each chapter builds on the previous one, apart from Chapter 5 which presents a general abstract framework to bridge the more classical expanding and hyperbolic systems explored in Chapters 3 and 4 with the nonuniformly expanding and partially hyperbolic systems described in Chapters 6 and 7. Throughout the book, the theory is illustrated with applications. A clear and detailed account of topics of current research interest, this monograph will be of interest to researchers in dynamical systems and ergodic theory. In particular, beginning researchers and graduate students will appreciate the accessible, self-contained presentation.
Book Synopsis Mathematics in the 21st Century by : Pierre Cartier
Download or read book Mathematics in the 21st Century written by Pierre Cartier and published by Springer. This book was released on 2014-11-15 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerous well-presented and important papers from the conference are gathered in the proceedings for the purpose of pointing directions for useful future research in diverse areas of mathematics including algebraic geometry, analysis, commutative algebra, complex analysis, discrete mathematics, dynamical systems, number theory and topology. Several papers on computational and applied mathematics such as wavelet analysis, quantum mechanics, piecewise linear modeling, cosmological models of super symmetry, fluid dynamics, interpolation theory, optimization, ergodic theory and games theory are also presented.
Book Synopsis Thermodynamic Formalism and Applications to Dimension Theory by : Luis Barreira
Download or read book Thermodynamic Formalism and Applications to Dimension Theory written by Luis Barreira and published by Springer Science & Business Media. This book was released on 2011-08-24 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained monograph presents a unified exposition of the thermodynamic formalism and some of its main extensions, with emphasis on the relation to dimension theory and multifractal analysis of dynamical systems. In particular, the book considers three different flavors of the thermodynamic formalism, namely nonadditive, subadditive, and almost additive, and provides a detailed discussion of some of the most significant results in the area, some of them quite recent. It also includes a discussion of the most substantial applications of these flavors of the thermodynamic formalism to dimension theory and multifractal analysis of dynamical systems.
Book Synopsis A Dynamical Systems Theory of Thermodynamics by : Wassim M. Haddad
Download or read book A Dynamical Systems Theory of Thermodynamics written by Wassim M. Haddad and published by Princeton University Press. This book was released on 2019-06-04 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: A brand-new conceptual look at dynamical thermodynamics This book merges the two universalisms of thermodynamics and dynamical systems theory in a single compendium, with the latter providing an ideal language for the former, to develop a new and unique framework for dynamical thermodynamics. In particular, the book uses system-theoretic ideas to bring coherence, clarity, and precision to an important and poorly understood classical area of science. The dynamical systems formalism captures all of the key aspects of thermodynamics, including its fundamental laws, while providing a mathematically rigorous formulation for thermodynamical systems out of equilibrium by unifying the theory of mechanics with that of classical thermodynamics. This book includes topics on nonequilibrium irreversible thermodynamics, Boltzmann thermodynamics, mass-action kinetics and chemical reactions, finite-time thermodynamics, thermodynamic critical phenomena with continuous and discontinuous phase transitions, information theory, continuum and stochastic thermodynamics, and relativistic thermodynamics. A Dynamical Systems Theory of Thermodynamics develops a postmodern theory of thermodynamics as part of mathematical dynamical systems theory. The book establishes a clear nexus between thermodynamic irreversibility, the second law of thermodynamics, and the arrow of time to further unify discreteness and continuity, indeterminism and determinism, and quantum mechanics and general relativity in the pursuit of understanding the most fundamental property of the universe—the entropic arrow of time.
Book Synopsis Introduction to the Modern Theory of Dynamical Systems by : Anatole Katok
Download or read book Introduction to the Modern Theory of Dynamical Systems written by Anatole Katok and published by Cambridge University Press. This book was released on 1995 with total page 828 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.
Book Synopsis Handbook of Dynamical Systems by : B. Hasselblatt
Download or read book Handbook of Dynamical Systems written by B. Hasselblatt and published by Elsevier. This book was released on 2002-08-20 with total page 1231 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volumes 1A and 1B. These volumes give a comprehensive survey of dynamics written by specialists in the various subfields of dynamical systems. The presentation attains coherence through a major introductory survey by the editors that organizes the entire subject, and by ample cross-references between individual surveys. The volumes are a valuable resource for dynamicists seeking to acquaint themselves with other specialties in the field, and to mathematicians active in other branches of mathematics who wish to learn about contemporary ideas and results dynamics. Assuming only general mathematical knowledge the surveys lead the reader towards the current state of research in dynamics. Volume 1B will appear 2005.
Book Synopsis Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by : Luís Barreira
Download or read book Ergodic Theory, Hyperbolic Dynamics and Dimension Theory written by Luís Barreira and published by Springer Science & Business Media. This book was released on 2012-04-28 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the last two decades, the dimension theory of dynamical systems has progressively developed into an independent and extremely active field of research. The main aim of this volume is to offer a unified, self-contained introduction to the interplay of these three main areas of research: ergodic theory, hyperbolic dynamics, and dimension theory. It starts with the basic notions of the first two topics and ends with a sufficiently high-level introduction to the third. Furthermore, it includes an introduction to the thermodynamic formalism, which is an important tool in dimension theory. The volume is primarily intended for graduate students interested in dynamical systems, as well as researchers in other areas who wish to learn about ergodic theory, thermodynamic formalism, or dimension theory of hyperbolic dynamics at an intermediate level in a sufficiently detailed manner. In particular, it can be used as a basis for graduate courses on any of these three subjects. The text can also be used for self-study: it is self-contained, and with the exception of some well-known basic facts from other areas, all statements include detailed proofs.
Book Synopsis Mathematics of Complexity and Dynamical Systems by : Robert A. Meyers
Download or read book Mathematics of Complexity and Dynamical Systems written by Robert A. Meyers and published by Springer Science & Business Media. This book was released on 2011-10-05 with total page 1885 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
Book Synopsis Thermodynamic Formalism, Statistical Properties and Multifractal Analysis of Non-uniformly Hyperbolic Systems by : Tianyu Wang (Ph. D. in mathematics)
Download or read book Thermodynamic Formalism, Statistical Properties and Multifractal Analysis of Non-uniformly Hyperbolic Systems written by Tianyu Wang (Ph. D. in mathematics) and published by . This book was released on 2021 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we concentrate on the study of different properties of non-uniformly hyperbolic systems. From section 1 to section 3, we introduce our main results, preliminaries and main techniques. The core of the thesis is in section 4 - section 6, where we illustrate the detailed proofs of our main results. The first result is on thermodynamic formalism. In section 4, we work with a C^1 non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Holder continuous potential with one additional condition, or geometric t-potential with t
